Maximal Element of Nest is Greatest Element

Theorem

Let $A$ be a nest.

Let $x$ be a maximal element of $A$.

Then $x$ is the greatest element of $A$.

Proof

Let $x$ be a maximal element of $A$.

Let $y \in A$ be arbitrary such that $x \ne y$.

We have by definition of maximal element that:

$x \not \subset y$

and as $x \ne y$:

$x \nsubseteq y$

But because $A$ is a nest:

$x \subseteq y$ or $y \subseteq x$

from which it follows that it must be the case that:

$y \subseteq x$

As $y$ is arbitrary, the result follows by definition of greatest element.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {II}$ -- Maximal principles: $\S 5$ Maximal principles